Interestingly, \(f^\omega (0)\)  is a complete shuffle of \((02)^\omega \)  and the paper folding sequence
\(1131133111331331\cdots \)  (after renaming 0 to 3); see Davis and Knuth [1]}
or Allouche and Shallit [2]}.
The latter claim was also verified by Shallit using Walnut, or by
considering the substitution rules for the paper folding sequence:
\(11 \mapsto 1101\,, \ 01 \mapsto 1001\,, \ 10 \mapsto 1100\,, \ 00 \mapsto 1000\,.\) 
